Young–Laplace equation

In physics, the Young–Laplace equation (/ləˈplɑːs/) is a nonlinear partial differential equation that describes the capillary pressure difference sustained across the interface between two static fluids, such as water and air, due to the phenomenon of surface tension or wall tension, although usage on the latter is only applicable if assuming that the wall is very thin. The Young–Laplace equation relates the pressure difference to the shape of the surface or wall and it is fundamentally important in the study of static capillary surfaces. It is a statement of normal stress balance for static fluids meeting at an interface, where the interface is treated as a surface (zero thickness): In physics, the Young–Laplace equation (/ləˈplɑːs/) is a nonlinear partial differential equation that describes the capillary pressure difference sustained across the interface between two static fluids, such as water and air, due to the phenomenon of surface tension or wall tension, although usage on the latter is only applicable if assuming that the wall is very thin. The Young–Laplace equation relates the pressure difference to the shape of the surface or wall and it is fundamentally important in the study of static capillary surfaces. It is a statement of normal stress balance for static fluids meeting at an interface, where the interface is treated as a surface (zero thickness): where Δ p {displaystyle Delta p} is the Laplace pressure, the pressure difference across the fluid interface, γ {displaystyle gamma } is the surface tension (or wall tension), n ^ {displaystyle {hat {n}}} is the unit normal pointing out of the surface, H {displaystyle H} is the mean curvature, and R 1 {displaystyle R_{1}} and R 2 {displaystyle R_{2}} are the principal radii of curvature. (Some authors refer inappropriately to the factor 2 H {displaystyle 2H} as the total curvature.) Note that only normal stress is considered, this is because it can be shown that a static interface is possible only in the absence of tangential stress. The equation is named after Thomas Young, who developed the qualitative theory of surface tension in 1805, and Pierre-Simon Laplace who completed the mathematical description in the following year. It is sometimes also called the Young–Laplace–Gauss equation, as Carl Friedrich Gauss unified the work of Young and Laplace in 1830, deriving both the differential equation and boundary conditions using Johann Bernoulli's virtual work principles. If the pressure difference is zero, as in a soap film without gravity, the interface will assume the shape of a minimal surface. The equation also explains the energy required to create an emulsion. To form the small, highly curved droplets of an emulsion, extra energy is required to overcome the large pressure that results from their small radius. In a sufficiently narrow (i.e., low Bond number) tube of circular cross-section (radius a), the interface between two fluids forms a meniscus that is a portion of the surface of a sphere with radius R. The pressure jump across this surface is related to the radius and the surface tension γ by This may be shown by writing the Young–Laplace equation in spherical form with a contact angle boundary condition and also a prescribed height boundary condition at, say, the bottom of the meniscus. The solution is a portion of a sphere, and the solution will exist only for the pressure difference shown above. This is significant because there isn't another equation or law to specify the pressure difference; existence of solution for one specific value of the pressure difference prescribes it. The radius of the sphere will be a function only of the contact angle, θ, which in turn depends on the exact properties of the fluids and the solids in which they are in contact: